
Journal of Convex Analysis 24 (2017), No. 1, 067073 Copyright Heldermann Verlag 2017 A Universal Bound on the Variations of Bounded Convex Functions Joon Kwon Institut de Mathématiques, Équipe Combinatoire et Optimisation, Université PierreetMarieCurie, 4 place Jussieu, 75252 Paris Cedex 05, France joon.kwon@enslyon.org [Abstractpdf] Given a convex set $C$ in a real vector space $E$ and two points $x,y\in C$, we investivate which are the possible values for the variation $f(y)f(x)$, where $f:C\longrightarrow [m,M]$ is a bounded convex function. We then rewrite the bounds in terms of the Funk weak metric, which will imply that a bounded convex function is Lipschitzcontinuous with respect to the Thompson and Hilbert metrics. The bounds are also proved to be optimal. We also exhibit the maximal subdifferential of a bounded convex function at a given point $x\in C$. Keywords: Convex functions, variations, Funk metric, Thompson metric, Hilbert metric. MSC: 26B25, 52A05 [ Fulltextpdf (103 KB)] for subscribers only. 